The undirected repetition threshold

For rational $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx’$, where $x$ is nonempty, $x’\in{x,x^\mathrm{R}}$, and $|xyx’|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mathrm{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $\mathrm{URT}(3)=\tfrac{7}{4}$. Then we show that $\mathrm{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that $\mathrm{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm this conjecture for $k\in{4,8,12}.$

💡 Research Summary

The paper introduces a new notion of “undirected r‑powers” in combinatorics on words, extending the classical concepts of ordinary r‑powers (words of the form xyx) and Abelian r‑powers (xy˜x) by allowing the third block to be either the original block x or its reversal xᴿ. Formally, an undirected r‑power is a word xyx′ where x is non‑empty, x′∈{x, xᴿ}, and the exponent r equals |xyx′|/|xy|. This definition corresponds to r‑powers up to the equivalence relation ≃ that identifies a word with its reversal. Consequently, for any alphabet size k we have the chain of inequalities RT(k) ≤ URT(k) ≤ ART(k), where RT(k) is the classical repetition threshold, URT(k) the newly defined undirected repetition threshold, and ART(k) the Abelian repetition threshold.

The authors first settle the case k = 3. Since Dejean proved RT(3)=7/4, the lower bound URT(3) ≥ 7/4 follows immediately. To obtain the matching upper bound, they construct a 24‑uniform morphism f and consider the infinite ternary word w = f^ω(0). Using exhaustive computer checks (assisted by the Walnut theorem‑proving system) they verify that w contains no factor of the form xyxᴿ with |x| > 3|y|, i.e., no undirected 7/4‑power. They also show that w is ordinary 7/4‑free, which together yields URT(3)=7/4. This demonstrates that for three letters the undirected threshold coincides with the classical one.

Next, the paper establishes a universal lower bound for all k ≥ 4: URT(k) ≥ (k‑1)/(k‑2). The proof proceeds in two parts. For k = 4,5 a computer‑backtracking search confirms that the longest undirected (k‑1)/(k‑2)‑free word has length k+3. For larger k, a combinatorial argument shows that any word of length k+4 must contain two equal letters separated by fewer than k‑2 other letters, which forces a factor of exponent at least (k‑1)/(k‑2). The authors illustrate this by constructing a decision tree (Figure 1) that enumerates all possible extensions of a prefix containing k‑1 distinct letters; each leaf of the tree inevitably yields a forbidden undirected power, confirming the bound.

The central conjecture (Conjecture 2) posits that the lower bound is tight for every k ≥ 4, i.e., URT(k) = (k‑1)/(k‑2). The authors verify this conjecture for the specific values k = 4, 8, 12. Their method adapts the encoding technique originally introduced by Pansiot. They define a morphism h that maps a smaller alphabet to blocks over Σ_k in such a way that the image of any (k‑1)/(k‑2)‑free word remains undirected (k‑1)/(k‑2)‑free. By carefully arranging the blocks (essentially grouping the alphabet into triples and interleaving them), they construct infinite words over 4‑, 8‑, and 12‑letter alphabets that avoid all undirected powers of exponent ≥(k‑1)/(k‑2). Since the lower bound already holds, this yields equality for those k.

Beyond repetition thresholds, the paper explores the broader context of pattern avoidance up to the equivalence ≃. It proves that a pattern is avoidable in the ordinary sense if and only if it is avoidable up to ≃, using the same direct‑product construction u⊗(123)^ω. As an illustration, the authors determine the avoidability index λ_≃(x^k): it equals 3 for k = 2,3 and 2 for all k ≥ 4, matching known results for Abelian patterns. They also note that any pattern avoidable up to ≃ remains avoidable after decorating its variables with reversals, linking undirected avoidance to the study of patterns with reversal.

In summary, the paper establishes the exact undirected repetition threshold for three letters, provides a general lower bound for all larger alphabets, confirms the conjectured exact value for several families of alphabet sizes, and connects these results to the theory of pattern avoidance. The work bridges ordinary, Abelian, and undirected repetitions, offering new tools for constructing long repetition‑free words and suggesting further research directions such as proving the conjecture for all k, investigating algorithmic aspects of undirected pattern detection, and applying the concepts to biological sequence analysis and coding theory.